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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Repeated Roots</dfn>(a) A real root, say <span class="process-math">\(r_1\)</span> is repeated <span class="process-math">\(s\)</span> times, then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
e^{r_1 x}, \quad x e^{r_1 x}, \cdots, x^{s-1} e^{r_1 x}
\end{equation*}
</div>
<p class="continuation">are the <span class="process-math">\(s\)</span> solutions.(b) A complex root, say <span class="process-math">\(\lambda+i\mu\)</span> is repeated <span class="process-math">\(s\)</span> times (it implies that <span class="process-math">\(\lambda-i\mu\)</span> is also a <span class="process-math">\(s\)</span> time repeated root), then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
e^{\lambda x} \cos \mu x, \quad e^{\lambda x} \sin \mu x, \quad x e^{\lambda x} \cos \mu x,  \quad x e^{\lambda x} \sin \mu x,\cdots, x^{s-1} e^{\lambda x} \cos \mu x, \quad x^{s-1} e^{\lambda x} \sin \mu x
\end{equation*}
</div>
<p class="continuation">are the <span class="process-math">\(2 s\)</span> solutions.</p>
<span class="incontext"><a href="sec4_2.html#p-159" class="internal">in-context</a></span>
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